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Question
Integrate the function:
`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`
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Solution
Let `1/(xsqrt(ax - x^2))`
Put `x = a/t`
dx = `- a/t^2 dt`
Now, `xsqrt(ax - x^2) = a/tsqrt(a xx a/t - a^2/t^2)`
`= a^2/t sqrt(1/t - 1/t^2) = a^2/t^2 sqrt(t - 1)`
`therefore I = 1/(a^2/t^2 sqrt(t - 1)) xx (- a)/t^2 dt`
`= - 1/a int 1/sqrt(t - 1) dt`
`= - 1/a ((t - 1)^(- 1/2 + 1))/(- 1/2 + 1) + C`
`= - 1/a (t - 1)^(1/2)/(1/2) + C`
`= - 2/a sqrt(t - 1) + C`
`= - 2/a sqrt(a/x - 1) + C`
`= - 2/a sqrt((a - x)/x) + C`
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